Trigonometry Polar Form of Complex Numbers

Complex numbers can be written in rectangular form z = x + yi, representing the rectangular coordinates of the point. They can also be written in the polar form, representing the polar coordinates of the point (using the distance from the origin and the angle it makes with the positive x-axis). To convert between the two forms, we use the formulas for the conversions. A formula that is very useful here and worth remembering is the Euler's formula for the conversion. In the lecture example questions, you'll also learn how to multiply two complex numbers in polar form.

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Hiï¼ŒProfessor At 10:48 you said -pi/4 is a positive term so we don't have to add pi to it, but it seems to me that -pi/4 is a negative term, wold you explain why?Thank you

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Last reply by: Dr. William MurraySun Apr 14, 2013 6:40 PM

Post by enya zhon April 12, 2013

I don't understand exactly what 'e' stands for in the formula re^theta*i. Is it the natural number? Please help me understand.Thanks!

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Last reply by: Dr. William MurraySun Apr 14, 2013 6:40 PM

Post by Dave Sealeon April 6, 2013

To further help me visualize the concepts of complex numbers in rectangular and polar form can you provide a few word problems and a couple real world applications where we can utilize these conversions please. I found it extremely helpful to grasp the concepts we are practicing from earlier lectures once I went through the real world applications for trigonometry lecture. Also, these conversions look like something that will show up in a tricky word problem in class, if you could walk us through 1 or 2 that would be fantastic. The word problems give me a sense of when I will need to go straight to conversions in the real life instances as well, I think they kind of go hand in hand and really solidify a strong understanding of these neat formulas and conversions. Thanks for all the help Dr. Murray, you do a great job of explaining advanced math concepts and I feel very prepared to take on AP trig next semester!

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Last reply by: Dr. William MurraySun Dec 9, 2012 7:36 AM

Post by valtteri viinikainenon December 9, 2012

4. (a) Given v = âˆ’3i and w = 1+ i express the product v^3 times w^2 in polar form and exponential form and show it is real.

(b) Find the solutions of z^3 = 1âˆ’ iand sketch the roots in the complex plane.

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Last reply by: Dr. William MurraySun Dec 9, 2012 7:41 AM

Post by varsha sharmaon June 9, 2011

For extra example 2 , we could have used pascals triangle for binomial expansion. that was another way to do.But converting to polar form was fun and i learnt it.Thanks.Varsha

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Polar Form of Complex Numbers

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

Each one of those was a pretty straight forward application of the formulas, one we had r=square root of (x2 + y2).0312

And then the arctan formula for (theta), remembering that you put a correction if the (x) is less than 0, and we found our (r), we found our (theta), we did put on the correction.0321

By the way,I have been doing all these in terms of radians, if you found arctan(1) in terms of degrees, if your calculator was in degree form it would have given you 45 and you would have to correct that in radians.0334

In this case, I did not even use a calculator because arctan(1) is a common value, I remember that was pi/4.0349

Add on my correction term of pi and I get 5pi/4 and so (z)=re(i)(theta) so 2 x 5pi/4(i).0358

On the other one we have to convert from polar to rectangular form, the polar form was 6e5pi/6(i).0370

You could use the conversion formula x=arcos(theta), y=arcsin(theta) or you can use the formula e(i)(theta) = cos(theta) + (i)sin(theta). 0378

I really like that one so I plugged that one in, drew a unit circle to remind me where 5pi/6 is and what is sin and cos are, fill those in and I got the rectangular coordinates for that complex number.0389

Now that would be a really nasty one if we had to multiply all that out to the 7th power.0005

Instead, what we are going to do is convert to polar form and hopefully the exponentiation will be easier in polar form and after we expand it out in polar form we will convert it back to rectangular form.0009

Let us see how it goes, remember r=square root(x2 + y2) and (theta) = arctan(y/x).0022

Sometimes you have to add on an extra pi there, you do that when (x) is less than 0.0036

1 + (i) that means that x is 1 and my y is one, r = square root (1+1) which is square root(2).0043

(theta) is arctan(1), that is a common value pi/4 and I do not have to introduce the fudge factor this time because the x is positive.0059

You can check that on the unit circle 1 + (i) is right there and that does check that the radius is the square root of 2 and the angle is pi/4, that checks my work here.0074

What we have here is square root of 2 x epi/4(i) that is the polar form of the complex number 1 + (i).0096

We want to raise that to the 7th power, so we raise both sides to the 7th power.0117

Now that looks pretty horrible but this just turns into square root of 27, now pi 4(i).0123

e to 1 power raised to another power, you just multiply the exponents, that just turns into e7th(i)/4(i) .0134

That is the beauty of the polar form is the exponents just multiply or add instead of making it really difficult in multiplying lots of things together.0148

The (e) part is already raised to the 7th power, (root 2)7 might take a little bit of work.0158

Let me look at this, I will write that as 21/2 to the 7th power and then you could write that as 27 ½.0165

If they're very rusty, you might want to go back and practice those formulas for converting a point into polar coordinates and back, because they'll be really helpful in this section of polar forms of complex numbers.0037